We propose a new approach to probe the spatial extension of the valence neutron orbital in the $^{9}$Be nucleus via the ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction. This property of the nuclear molecular orbital has not been established in previous experimental studies and divergence exists between the theoretical descriptions of ${}^{9}$Be from different perspectives, textit{i.e.}, the antisymmetrized molecular dynamics and the container pictures of cluster dynamics. These pictures are represented by two different well-proven microscopic models, the antisymmetrized molecular dynamics (AMD) and Tohsaki-Horiuchi-Schuck-R{o}pke (THSR) wave functions. The corresponding reduced width amplitudes (RWAs) in the $^{8}$Be$+n$ channel are extracted from both the AMD and THSR wave functions, and they are found to describe drastically different valence-nucleon motion, which shows the theoretical ambiguity in describing the $pi$-orbitals in $^{9}$Be. Using the RWAs as input, the physical observables of the ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction are predicted by the distorted-wave impulse approximation (DWIA) framework. The magnitudes of the triple-differential cross sections (TDX) are found to be highly sensitive to the RWA input. It is concluded that the ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction could provide a feasible probing for the subtle differences between several structure models manifesting through the spatial extension of the $pi$-orbital in the $^{9}$Be nucleus.
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